EM field and wave interactions of a point charge

I've been thinking of 2 point charges separated by some distance in static equilibrium. When one charge is moved from rest, the EM field would change the way it looks at the location of the other point charge. This "changing in the looks" of the EM field as I understand propagates from the first charge at the speed of light and constitutes an EM wave. This is easy to see using a computer simulation that allows dragging charges around. When the change in the EM field reaches the other point charge it accelerates, trying to restore equilibrium.

So, I know about cases where an excited atom returns to its ground state and how to calculate the frequency and wavelength of the emitted photon. I'm wondering how I might do that with the somewhat unreal situation above. Could I just consider the work done on the second charge to be equal to the light's energy, then divide by Planck's constant to get frequency (if both charges had charge e)? Thanks.

The nature of the light will depend on how the charges move. In such general situation as you described, the motion would be most probably aperiodic, so the light will move in all directions in a complicated way. One cannot assign frequency to it in such situation (one can resolve it into Fourier components, but the result would be complicated as well.)

Frequency is a quantity which refers to periodic wave, like sine wave. In order to get such wave, the charges have to move periodically - in circles in synchrotron, or around each other, like in excited state of hydrogen atom, or oscillate rectilinearly as in antenna.

No problem. From your post, it seems you learned bit about quanta. The Planck constant and the connection between the energy and frequency is important, but rather subtle and not entirely clear, so I recommend to learn bit about the classical theory of light (wave optics/electrodynamics ) which is quite clear and natural, and only then move on to learn about the quantum theory.